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  • THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC
Title: THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC
Abstract Let $K$ be an algebraically closed field of prime characteristic $p$ , let $X$ be a semiabelian variety defined over a finite subfield of $K$ , let $\unicode[STIX]{x1D6F7}:X\longrightarrow X$ be a regular self-map defined over $K$ , let $V\subset X$ be a subvariety defined over $K$ , and let $\unicode[STIX]{x1D6FC}\in X(K)$ . The dynamical Mordell–Lang conjecture in characteristic $p$ predicts that the set $S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$ is a union of finitely many arithmetic progressions, along with finitely many $p$ -sets, which are sets of the form $\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$ for some $m\in \mathbb{N}$ , some rational numbers $c_{i}$ and some non-negative integers $k_{i}$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $X$ is an algebraic torus, we can prove the conjecture in two cases: either when $\dim (V)\leqslant 2$ , or when no iterate of $\unicode[STIX]{x1D6F7}$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $X$ . We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction.  more » « less
Award ID(s):
1646385 1800492
NSF-PAR ID:
10231641
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
Journal of the Institute of Mathematics of Jussieu
Volume:
20
Issue:
2
ISSN:
1474-7480
Page Range / eLocation ID:
669 to 698
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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